Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Is it true that, for every $\epsilon>0$, if $n$ is sufficiently large, every subgraph of $Q_n$ with $$\geq \epsilon n2^{n-1}$$ many edges contains a $C_6$?

Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges). Is it true that, for every $\epsilon>0$, if $n$ is sufficiently large, every subgraph of $Q_n$ with $$\geq \epsilon n2^{n-1}$$ many edges contains a $C_6$?

The answer to this problem is no: Chung [Ch92] and Brouwer, Dejter, and Thomassen [BDT93] constructed an edge-partition of $Q_n$ into four subgraphs, each containing no $C_6$.